Comparing Quantities Chapter Notes | Mathematics (Maths) Class 8 PDF Download

Recalling Ratios and Percentages

Ratio: A ratio is a way of comparing two or more quantities. For example, 2:3 is a ratio, which means for every two parts of one thing, there are three parts of another.
Percentages: A percentage is a number or ratio that can be expressed as a fraction of 100. 

Ratios are a way to show how two quantities relate to each other. Imagine you have 5 green marbles and 2 pink marbles. If you want to compare the number of green marbles to the number of pink marbles, you use a ratio. Similarly, percentages help us easily understand how big or small a part is compared to the whole. So, if you see a 50% discount on a toy, it means the price is cut in half!

RatiosLets understand this with the help of an example: 

A basket contain 20 apples and 5 oranges. 

1. Ratio of number of oranges to the number of apples = 5 : 20. (5 is to 20)
This means for every 5 oranges there exist 20 apples.
This can also be expressed as a fraction, which would be 5/20 or 1/4.
2. Percentage of apples =  (Total number of apples / Total number of fruits)x100
=> (20/25)x100
=> 80%

Let's dive deeper into this topic by solving a detailed problem.

If a percentage is in decimal like 50.69% then we will round off the number to the nearest integer.
For example 50.69%  = 51% if we round it off to the nearest tens.

Sales Tax/ Value Added Tax/ Goods and Services Tax

Imagine you're buying a toy that costs $100. But when you go to pay, the cashier asks for $110 instead of $100. Why? That's because of something called a sales tax or VAT.

Sales Tax/VAT is a little extra money you have to pay when you buy something. This extra money goes to the government so they can use it to build things like roads, schools, and hospitals. The percentage of the extra money can vary, so sometimes you pay a bit more or a bit less, depending on the kind of thing you are buying and from which area.

Total Bill = Actual Amount + Tax Amount

Example : (Finding Sales Tax) The cost of a pair of roller skates at a shop was ₹450. The sales tax charged was 5%. Find the bill amount.

 Sol: Total Bill = ₹450 + tax
Tax = 450x(5/100) = ₹22.5
Total Bill = 450 + 22.5 = ₹472.50

Compound Interest

Let's first remember what simple interest is. 

Simple interest is a way to calculate the extra money you earn or pay when you save or borrow money. It’s called "simple" because it’s easy to calculate and doesn’t change over time. 

Formula for Simple Interest:

where, 

• P= Principal is the original amount of money you save or borrow.
• R= Rate is the percentage of interest per year.
• T= Time is the number of years the money is saved or borrowed.

Example : A sum of  ₹10,000 is borrowed at a rate of interest 15% per annum for 2 years. Find the simple interest on this sum and the amount to be paid at the end of 2 years.
Sol: Interest Charged For One Year =  

Interest for 2 years = ₹ 1500 × 2 =  ₹3000 Amount to be paid at the end of 2 years = Principal + Interest =  ₹ 10000 +  ₹3000 =  ₹13000

Calculating Compound Interest

Now, let's understand compound interest. 

Compound interest is interest earned on the principal sum and previously accumulated interest. It's also known as "interest on interest".

The formula for compound interest is as follows:

CI = Final Amount - Initial Amount

Deducing a Formula for Compound Interest

To better our understanding of the concept, let us take a look at the compound interest formula derivation. Here we will take our principal to be Rupee.1/- and work our way towards the interest amounts of each year gradually.

Year 1

• The interest on Rupee 1/- for 1 year is equal to r/100 = i (assumed)

• Interest after Year 1 is equal to Pi

• FV (Final Value) after Year 1 is equal to P + Pi = P(1+i)

Year 2

• Interest for Year 2 = P(1+i) × i

• FV after year two is equal to P(1+i) + P(1+i) × i = P(1+i)²

Year ‘t’

• Final Value (Amount) after year “t” is equal to P(1+i)^t

• Now substituting actual values we get Final Value is equal to ( 1 + R/100)^t

• CI = FV – P is equal to P ( 1 + R/100)^t – P

This is how the formula of compound interest can be derived. 

Applications of Compound Interest Formula

Some of the applications of compound interest are:

1. Increase in population or decrease in population.

2. Growth of bacteria.

3. Rise in the value of an item.

4. Depreciation in the value of an item.

Example: The value of a piece of art was $5,000 in the year 2010. It increased in value at a rate of 4% per year. Find the value of the art at the end of the year 2015.

Sol: To find the value of the art at the end of the year 2015, we use the compound interest formula:

$A=P(1+\frac{r}{100})t$

• Principal ($PP$) = $5,000
• Rate ($rr$) = 4%
• Time ($tt$) = 2015 - 2010 = 5 years

$A=5000(1+\frac{4}{100})5$

$A=5000(1+0.04)5A\; =\; 5000\; \backslash left(1\; +\; 0.04\backslash right)^5$ 

$A=5000\left(1.04\right)5A\; =\; 5000\; \backslash left(1.04\backslash right)^5$

≈1.2167

Multiply by the principal:

$A=5000\times 1.2167\approx 6083.50A\; =\; 5000\; \backslash times\; 1.2167\; \backslash approx\; 6083.50$

So, the value of the art at the end of the year 2015 is approximately $6,083.50.